One could mimic the proof of the standard local limit theorem, but we prefer a shorter and less technical approach based on [1]}.
We first need to check that a central limit theorem holds. Recall (see e.g. [2]}) that if \(Y\)  is a random variable with a finite third moment, then
\(\mathopen {}\mathclose {\left|\mathbb {E}\mathopen {}\mathclose {\left[\operatorname{e}^{{i\hspace{0.55542pt}}Y}\right]} - \mathopen {}\mathclose {\left(1 + {i\hspace{0.55542pt}}\mathbb {E}[Y] - \frac{1}{2} \mathbb {E}[Y^{2}]\right)}\right|} \leqslant \frac{1}{6} \mathbb {E}[|Y|^{3}].\) 
